"Mind if I ask you some deep questions?"
"Uhh..."
"What is the greatest depth at which sea creatures are known to live?"
"Huh?"
"How far down does the largest mine shaft go?"
"Umm... What?"
"Can you name a Jules Verne novel?"

Just ask her "What is the meaning of life?", or if she knows the tale of the edmund fitzgerald.

Ask her if she ever thinks about the potential long term impact our species uncontrolled population will have on our planet is positive or negative?

Why do you want to ask these questions anyway?

__________________

Of all the things I have seen, the realization that the grim reaper laughs at me has been my greatest motivation to succeed.A mask is not a symbol of a lie but a symbol of adaptationCats are wonderful creatures, and unlike humans they love you for who you are.

SteinerOfThule has a very good point, deep discussions don't necessarily just spawn from single questions, how about just letting the conversation flow, if it does that, and then try directing it towards deeper or more profound topics.
I got an idea; ask about music, and then how it makes her feel or something. Don't just be asking questions though I guess it'd be optimal to have something you can offer to the conversation as well.